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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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All sourcesNASA NTRSDOE OSTI
NASA NTRS1993

A SIMULINK Environment for Flight Dynamics and Control Analysis - Application to the DHC-2 'Beaver', Part 1. Implementation of a Model Library in SIMULINK

The design of advanced Automatic Aircraft Control Systems (AACS's) can be improved upon considerably if the designer can access all models and tools required for control system design and analysis through a graphical user-interface, from within one software environment. This MSc-thesis presents the first step in the development of such an environment, which is currently being done at the Section for Stability and Control of Delft University of Technology, Faculty of Aerospace Engineering. The environment is implemented within the commercially available software package MATLAB/SIMULINK. The report consists of two parts. Part I gives a detailed description of the AACS design environment. The heart of this environment is formed by the SIMULINK implementation of a nonlinear aircraft model in block-diagram format. The model has been worked out for the old laboratory aircraft of the Faculty, the De Havilland DHC-2 'Beaver', but due to its modular structure, it can easily be adapted for other aircraft. Part I also describes MATLAB programs which can be applied for finding steady-state trimmed-flight conditions and for linearization of the aircraft model, and it shows how the built-in simulation routines of SIMULINK have been used for open-loop analysis of the aircraft dynamics. Apart from the implementation of the models and tools, a thorough treatment of the theoretical backgrounds is presented. Part II of this report presents a part of an autopilot design process for the 'Beaver' aircraft, which clearly demonstrates the power and flexibility of the AACS design environment from part I. Evaluations of all longitudinal and lateral control laws by means of nonlinear simulations are treated in detail. The AACS design environment from part I proved to be a very useful tool for designing the control laws of the 'Beaver' autopilot within a very tight time-schedule. The autopilot design process itself will be used as a guideline for future AACS research at the Faculty of Aerospace Engineering. Flight tests of the 'Beaver' autopilot, done after evaluating the control laws in the SIMULINK package, proved to be quite successful. In the future, the AACS design package will evolve into a standardized, integrated design environment which can be applied to virtually any type of aircraft. The AACS design cycle will be shortened further by developing tools for automatically porting control laws from the MATLAB/SIMULINK environment to a piloted real-time flight simulator and the Flight Control Computers of the aircraft.

Aircraft Modelling↗
DOE OSTI2023

Enhancing Charge Storage of Mo 2 Ti 2 C 3 MXene by Partial Oxidation

Abstract Driving the pseudocapacitive redox intercalation in 2DMXenes with neutral electrolytes is important for safer, more sustainable, and improved electrochemical charge storage. Single transition metal MXenes, such as Ti 3 C 2 , have shown great promise for energy storage, owing to their high conductivity and redox activity. Mixed metallic MXenes, such as out‐of‐plane ordered Mo 2 Ti 2 C 3 , have remained underexplored in energy storage because of the absence of redox activity in most of the electrolytes. Simultaneous structural modifications and instigating intercalation pseudocapacitance in neutral electrolytes could be a viable strategy for enhancing their electrochemical properties. Herein, a facile synthesis of partially oxidized Mo 2 Ti 2 C 3 MXene (PO‐Mo 2 Ti 2 C 3 ) exhibiting improved charge storage capability is demonstrated. Optical, structural, and spectroscopic analyses indicate the formation of oxide nanostructures upon thermal oxidation of Mo 2 Ti 2 C 3 . This leads to an enhanced energy storage capability with remarkably improved cyclability as well as a high Coulombic efficiency in a neutral LiCl electrolyte. This work highlights the importance of structural modifications of MXenes to enhance their charge storage and shows the promise of less explored double transition metal MXenes in energy storage.

Chemistry↗
DOE OSTI2023

Smart Droplets Stabilized by Designer Surfactants: From Biomimicry to Active Motion to Materials Healing

The science and technologies of emulsion droplets have been a long‐term focus of extensive research endeavors for their practical utility across a breadth of industries, including pharmaceutical products, oil recovery processes, and the food sciences. However, with advances in materials chemistry and characterization tools, new emerging areas are arising with a focus on “smart droplets”. The versatility of emulsion droplets across is based on their ability to partition and create isolated systems with properties defined by the liquid–liquid interface, while preparative routes allow manipulation of droplet size, stability, and encapsulated contents. As described in this article, significant efforts are being devoted to creating new types of droplets by “activating” this interface through the incorporation of reactive structures that trigger droplet response to applied or environmental stimuli (e.g., pH, temperature, salt, or external fields). Moreover, parallels between droplets and live cells inspire efforts to conceive systems that resemble biological motifs or that can produce cellular behaviors that imitate biology (e.g., swarming, communication, or motion). Here, the authors highlight recent advances in smart droplets, with emphasis on organic, polymer, and/or particle surfactants that give rise to inter‐droplet communication (via aggregation, fusion, division, or mass transfer), droplet vehicles for controlled delivery, autonomous droplet motion, and tunable emulsion inversion. Especially emphasized is the macromolecular design to produce reactive and functional surfactants, which are crucial to responsive droplet behavior and their underlying mechanisms. More generally, the exquisite interplay between materials science and biology inspires the review of this research area that provides unique opportunities for insight and inspiration into the capabilities of new droplet designs.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Effect of Preparation Conditions of Fe@SiO2 Catalyst on Its Structure Using High-Pressure Activity Studies in a 3D-Printed SS Microreactor

Fischer–Tropsch synthesis (FTS) in a 3D-printed stainless steel (SS) microchannel microreactor was investigated using Fe@SiO2 catalysts. The catalysts were prepared by two different techniques: one pot (OP) and autoclave (AC). The mesoporous structure of the two catalysts, Fe@SiO2 (OP) and Fe@SiO2 (AC), ensured a large contact area between the reactants and the catalyst. They were characterized by N2 physisorption, H2 temperature-programmed reduction (H2-TPR), scanning electron microscopy (SEM), transmission electron microscopy (TEM), X-ray diffraction (XRD), Fourier-transform infrared spectroscopy (FTIR), X-ray photoelectron microscopy (XPS), and thermogravimetric analysis–differential scanning calorimetry (TGA-DSC) techniques. The AC catalyst had a clear core–shell structure and showed a much greater surface area than that prepared by the OP method. The activities of the catalysts in terms of FTS were studied in the 200–350 °C temperature range at 20-bar pressure with a H2/CO molar ratio of 2:1. The Fe@SiO2 (AC) catalyst showed higher selectivity and higher CO conversion to olefins than Fe@SiO2 (OP). Stability studies of both catalysts were carried out for 30 h at 320 °C at 20 bar with a feed gas molar ratio of 2:1. The Fe@SiO2 (AC) catalyst showed higher stability and yielded consistent CO conversion compared to the Fe@SiO2 (OP) catalyst.

Biochemistry & Molecular Biology↗
DOE OSTI2024

High-Density Capacitive Energy Storage in Low-Dielectric-Constant Polymer PMMA/2D Mica Nanofillers Heterostructure Composite

The ubiquitous, rising demand for energy storage devices with ultra-high storage capacity and efficiency has drawn tremendous research interest in developing energy storage devices. Dielectric polymers are one of the most suitable materials used to fabricate electrostatic capacitive energy storage devices with thin-film geometry with high power density. In this work, we studied the dielectric properties, electric polarization, and energy density of PMMA/2D Mica nanocomposite capacitors where stratified 2D nanofillers are interfaced between the multiple layers of PMMA thin films using two heterostructure designs of the capacitors, PMMA/2D Mica/PMMA (PMP) and PMMA/2D Mica/PMMA/2D Mica/PMMA (PMPMP). The incorporation of a 2D Mica nanofiller in the low-dielectric-constant PMMA leads to an enhancement in the dielectric constant, with ∆ε ~ 15% and 53% for PMP and PMPMP heterostructures at room temperature. Additionally, a significant improvement in discharged energy density was measured for the PMPMP capacitor (Ud ~ 38 J/cm3 at 825 MV/m) compared to the pristine PMMA (Ud ~ 9.5 J/cm3 at 522 MV/m) and PMP capacitors (Ud ~ 19 J/cm3 at 740 MV/m). This excellent capacitive and energy storage performance of the PMMA/2D Mica heterostructure nanocomposite may inform the fabrication of thin-film, high-density energy storage capacitor devices for potential applications in various platforms.

Biochemistry & Molecular Biology↗
DOE OSTI2024

Resonant Metasurfaces with Van Der Waals Hyperbolic Nanoantennas and Extreme Light Confinement

This work reports on a metasurface based on optical nanoantennas made of van der Waals material hexagonal boron nitride. The optical nanoantenna made of hyperbolic material was shown to support strong localized resonant modes stemming from the propagating high-k waves in the hyperbolic material. An analytical approach was used to determine the mode profile and type of cuboid nanoantenna resonances. An electric quadrupolar mode was demonstrated to be associated with a resonant magnetic response of the nanoantenna, which resembles the induction of resonant magnetic modes in high-refractive-index nanoantennas. The analytical model accurately predicts the modes of cuboid nanoantennas due to the strong boundary reflections of the high-k waves, a capability that does not extend to plasmonic or high-refractive-index nanoantennas, where the imperfect reflection and leakage of the mode from the cavity complicate the analysis. In the reported metasurface, excitations of the multipolar resonant modes are accompanied by directional scattering and a decrease in the metasurface reflectance to zero, which is manifested as the resonant Kerker effect. Van der Waals nanoantennas are envisioned to support localized resonances and can become an important functional element of metasurfaces and transdimensional photonic components. By designing efficient subwavelength scatterers with high-quality-factor resonances, this work demonstrates that this type of nanoantenna made of naturally occurring hyperbolic material is a viable substitute for plasmonic and all-dielectric nanoantennas in developing ultra-compact photonic components.

Chemistry↗
DOE OSTI2026

Imaging a terahertz superfluid plasmon in a two-dimensional superconductor

The superconducting gap defines the fundamental energy scale for the emergence of dissipationless transport and collective phenomena in a superconductor. In layered high-temperature cuprate superconductors, in which the Cooper pairs are confined to weakly coupled two-dimensional (2D) copper–oxygen (CuO 2 ) planes, terahertz (THz) spectroscopy at subgap millielectronvolt (meV) energies has provided crucial insights into the collective superfluid response perpendicular to the superconducting layers. However, within the CuO 2 planes, the collective superfluid response manifests as plasmonic charge oscillations at energies far exceeding the superconducting gap, obscured by strong dissipation. Here, in this study, we present spectroscopic evidence of a below-gap, 2D superfluid plasmon in few-layer Bi 2 Sr 2 CaCu 2 O 8+x and spatially resolve its deeply subdiffractive THz electrodynamics. By placing the superconductor in the near field of a spintronic THz emitter, we reveal this distinct resonance—absent in bulk samples and observed only in the superconducting phase—and determine its plasmonic nature by mapping the geometric anisotropy and dispersion. Crucially, these measurements offer a direct view of the momentum-dependent and frequency-dependent superconducting transition in two dimensions.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2024

Mechanically and Thermally Robust Gel Electrolytes Built from A Charged Double Helical Polymer

Polymer electrolytes have received tremendous interest in the development of solid-state batteries, but often fall short in one or more key properties required for practical applications. Herein, a rigid gel polymer electrolyte prepared by immobilizing a liquid mixture of a lithium salt and poly(ethylene glycol) dimethyl ether with only 8 wt% poly(2,2′-disulfonyl-4,4′-benzidine terephthalamide) (PBDT) is reported. The high charge density and rigid double helical structure of PBDT lead to formation of a nanofibrillar structure that endows this electrolyte with stronger mechanical properties, wider temperature window, and higher battery rate capability compared to all other poly(ethylene oxide) (PEO)-based electrolytes. The ion transport mechanism in this rigid polymer electrolyte is systematically studied using multiple complementary techniques. Li/LiFePO 4 cells show excellent capacity retention over long-term cycling, with thermal cycling reversibility between ambient temperature and elevated temperatures, demonstrating compelling potential for solid-state batteries targeting fast charging at high temperatures and slower discharging at ambient temperature.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Ultrafast low-temperature metal–insulator interface phonon dynamics and heat transport in a Pt/Gd 3 Fe 5 O 12 heterostructure

Interfacial thermal and acoustic phenomena have an important role in quantum science and technology, including in spintronic and spincaloritronic materials and devices. Simultaneous measurements of the low-temperature thermal and acoustic properties of a metal/insulator heterostructure reveal distinct dynamics in the characteristic phonon frequency ranges of acoustic and thermal transport. The measurements probed a heterostructure consisting of a thin film of Pt on the ferrimagnetic insulator gadolinium iron garnet (Gd 3 Fe 5 O 12 , GdIG) grown epitaxially on a gadolinium gallium garnet substrate. Ultrafast structural dynamics within the Pt layer were tracked using time-resolved ultrafast x-ray diffraction and analyzed to probe interfacial acoustic and thermal properties. The rapid heating of the Pt layer by a 400 nm wavelength femtosecond-duration optical pulse produced transient structural changes that provided the stimulus for these measurements. Rapid heating produced a broadband acoustic pulse that was partially reflected by the Pt/GdIG interface. Temporal frequencies up to 740 GHz, corresponding to angular frequencies of several THz, were detected in a wavelet analysis of the acoustic oscillations of the strain in the Pt layer. The structural results were analyzed to determine (i) the acoustic damping coefficient and phonon mean free path in Pt at frequencies of hundreds of GHz and (ii) the Grüneisen anharmonicity parameter. The thermal conductance of the Pt/GdIG interface was tracked using the slower, tens-of-picosecond-scale, dynamics of the initial cooling of the heated Pt layer. Analysis using a model based on the Boltzmann transport equation shows that the phonon transmission is lower at the phonon frequencies relevant to thermal transport than for subterahertz regime acoustics.

Acoustic phenomena↗
DOE OSTI2024

Technoeconomic and Emissions Analysis of the Hybrid Redox Process for the Production of Acetic Acid with CO 2 Utilization

The production of oxygenated hydrocarbons, such as acetic acid, using captured CO 2 is a promising pathway to reduce greenhouse gas emissions in the chemical industry. The use of a chemical looping‐based hybrid redox process (HRP) is proposed to convert CO 2 and natural gas into separate CO and syngas streams that can be used to produce various commodity oxygenates, while allowing the beneficial utilization of captured CO 2 . Here, a detailed technoeconomic analysis of HRP applied to the production of acetic acid is presented. Emissions and energy analyses show the ability of HRP to lower the CO 2 emissions for acetic acid synthesis by 74% compared to a conventional steam and autothermal reforming route. HRP also offers a potential 34% reduction in capital costs. Compared to a dry reforming based acetic acid production route, HRP has the potential for significantly lower costs. If integrated with a low carbon energy source, HRP has the potential to achieve a negative emission of greenhouse gas (‐0.50 kg CO 2 per kg acetic acid).

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

Computational modeling of coupled mechanical damage and electrochemistry in ternary oxide composite electrodes

Performance degradation of ternary layered oxide cathodes largely originates from their loss of structural integrity in cyclic usage. Mechanical damage, such as intergranular fracture of the active particles, is not only a mechanical cleavage process but also interferes with electrochemical kinetics such as infiltration of liquid electrolyte, surface corrosion of the constituent primary particles, and may eventually isolate the primary grains from the electron conducting network. Here, in this work, we develop a computational framework that integrates electrochemistry of a LiNi x Mn y Co 1−x−y O 2 (NMC) composite cathode with mechanical damage of the active particles. To fully examine the intricate chemomechanical behavior of the electrode, we evaluate the effects of the anisotropic material properties, the influence of mechanical potential on Li transport, and the concurrent intergranular fracture and electrolyte penetration along the grain boundaries upon multiple cycles. Electrolyte infiltration benefits capacity retention but aggravates further mechanical damage by corrosion. Structural failure mostly occurs in the first charging due to the anisotropic mechanical strain between the primary grains, while the resulting damage remains stable in the later few cycles. The results are consistent with experimental observations and the integration of electrochemistry and mechanical failure enables a step further understanding of the complex mechanism of battery degradation.

Battery degradation↗
DOE OSTI2026

Role of Amorphous Chains in Nanoplastic Formation from Semicrystalline Polymers

Semicrystalline polymers, e.g., polyethylene terephthalate (PET), release micro- (100 nm to 1 mm) and nanoplastics (up to 100 nm) [MNPL] when they are degraded under quiescent conditions. However, the exact molecular mechanisms leading to material fragmentation into MNPL are unknown. Here, we monitor the evolution of chain molecular weight and the MNPL production kinetics during hydrolysis of PET pellets. We find that only ~0.6% of the amorphous phase ester bonds are hydrolyzed at the onset of MNPL release. Then, by combining a random scission model with measured amorphous spacings, we estimate that only ~15% of the stress transmitters in the amorphous phase, namely, bridges and entangled loops, have failed by this point. Thus, spontaneous fragmentation of the semicrystalline nanostructure occurs despite significant intercrystalline connectivity. We resolve this apparent contradiction by proposing that fragmentation can only occur when progressive tie-chain scission causes the material to undergo the ductile/brittle transition. Mechanistically, we speculate that the internal stresses responsible for material fragmentation are caused by processing (i.e., residual stresses) and/or sample densification induced by chemi-crystallization. We discuss additional factors that may affect our analysis and thus require further investigation, such as skin-core effects, preferential degradation of stress transmitters and/or recrystallization processes.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Interlayer Design for Halide Electrolytes in All‐Solid‐State Lithium Metal Batteries

Abstract All‐solid‐state lithium‐metal batteries (ASSLMBs) are promising for transportation electrification due to their superior safety and high energy density. Lithium halide electrolytes provide excellent processing flexibility, high ionic conductivity, and anodic stability (>4.1 V), making them highly compatible with high‐voltage cathodes, surpassing sulfide electrolytes (<2.1 V). Nevertheless, halide electrolytes suffer from low cathodic stability and form an electronically conductive interphase with lithium, resulting in a critical current density (CCD) of nearly zero. Herein, Li 3 YbCl 6 electrolytes are synthesized that are kinetically stable with lithium by forming an electronic insulating solid electrolyte interphase. Guided by critical overpotential criteria, a PI 3 interlayer is designed that transforms into Li 6 PI 3 upon contact with lithium, substantially reducing the interfacial resistance of Li 3 YbCl 6 against lithium to 34 Ω and achieving a high critical overpotential of 114 mV. By substituting Yb with Lu, Li 3 LuCl 6 electrolytes with Li 6 PI 3 interlayers reach a CCD of 1.0 mA cm −2 at a capacity of 1.0 mAh cm −2 , comparable to sulfide electrolytes but with higher oxidation stability. Additionally, Li 6 PI 3 enables stable cycling of Li//Li cells with Li 3 LuCl 6 electrolytes at 0.5 mA cm −2 for 400 cycles and maintains 86.5% capacity in Li//LiCoO 2 cells after 220 cycles at 30 °C, paving the way for high‐performance ASSLMBs.

Chemistry↗
DOE OSTI2026

Few is different: deciphering many-body dynamics in mesoscopic quantum gases

Emergent macroscopic descriptions of matter, such as hydrodynamics, are central to our description of complex physical systems across a wide spectrum of energy scales. The conventional understanding of these many-body phenomena has recently been shaken by a number of experimental findings. Collective behavior of matter has been observed in mesoscopic systems, such as high-energy hadron–hadron collisions, or ultracold gases with only a few strongly interacting fermions. In such systems, the separation of scales between macroscopic and microscopic dynamics (at the heart of any effective theory) is inapplicable. To address the conceptual challenges that arise from these observations and explore the universality of emergent descriptions of matter, the EMMI Rapid Reaction Task Force was assembled. This document summarizes the RRTF discussions on recent theoretical and experimental advances in this rapidly developing field. Leveraging technological breakthroughs in the control of quantum systems, we can now quantitatively explore what it means for a system to exhibit behavior beyond the sum of its individual parts. In particular, the report highlights how the (in)applicability of hydrodynamics and other effective theories can be probed across three principal frontiers: the size frontier, the equilibrium frontier, and the interaction frontier.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2026

Stoichiometry dependent properties of cerium hydride: An active learning developed interatomic potential study

Cerium hydride has a variety of interesting properties, including a known lattice contraction and densification with increasing hydrogen content. However, precise stoichiometric control is not experimentally straightforward and ab initio approaches are not computationally feasible for many properties such as melting and low temperature diffusion. Therefore, we develop a machine-learned interatomic potential for cerium hydride that is valid for H to Ce ratios from 2.0 to 3.0. A query-by-committee active learning approach is used to develop the training set. Leveraging classical molecular dynamics simulations, we assess a range of properties and provide fundamental mechanisms for the trends with stoichiometry. Finally, a majority of the properties follow the trend of lattice contraction, being governed by the stronger lattice binding induced by adding octahedral atoms.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Mitigating Sodium Ordering for Enhanced Solid Solution Behavior in Layered NaNiO 2 Cathodes

The O-type layered nickel oxides suffer from undesired cooperative Jahn–Teller distortion stemming from Ni 3+ ions and undergo multiple biphasic structural transformations during the insertion/extraction of large Na + ions, posing a significant challenge to stabilize the structural integrity. We present here a systematic investigation of the impact of substituting 5 % divalent (Mg 2+ ) or trivalent (Al 3+ or Co 3+ ) ions for Ni 3+ to alleviate Na + ion ordering and perturb the Jahn–Teller effect to enhance structural stability. Here, we gauge a fundamental understanding of the Mg−O and Na−O or Mg−O−Na bonding interactions, noting that the ionicity of the Mg−O bond deshields the electronic cloud of oxygen from Na + ions. Furthermore, calculations of the Van Vleck distortion modes reveal a relaxation of NiO 6 octahedra from Jahn–Teller distortion and a reduced electron density at the interlayer with Mg 2+ substitution. Long-range (operando X-ray diffraction) and short-range (magic angle spinning nuclear magnetic resonance) structural analyses provide insights into reduced ordering, allowing a stable continuous solid solution. Overall, Mg-substitution results in a high-capacity retention of ~96 % even after 100 cycles, showcasing the potential of this strategy for overcoming the structural instabilities and enhancing the performance of sodium-ion batteries.

25 ENERGY STORAGE↗
DOE OSTI2025

Bi‐Continuous W‐Rich Refractory High Entropy Alloy‐Cu Composite: Toward Material Innovation of Nuclear Reactor Coolant System

Abstract Refractory high‐entropy alloys (RHEAs) are considered promising candidate materials for next‐generation nuclear reactors due to their superior mechanical strength, irradiation resistance, and thermal stability at high temperatures. However, the significant positive heat of mixing between refractory alloying elements and Cu, commonly used in cooling systems, poses challenges in forming composite structures. This study addresses the issue using a liquid metal dealloying (LMD) process. A precursor alloy (WTaVTi) with a directional dendrite‐interdendrite structure is fabricated and reacted with molten Cu at 1200 °C for 96 h. This approach produced a RHEA‐Cu composite with a stable interface between RHEA (W 31.5 Ta 30.9 V 21.4 Ti 14.3 ) and Cu, featuring a spontaneously formed W‐rich interlayer that enhances interfacial bonding. The composite showed excellent irradiation resistance, with 30% less swelling under α‐ion irradiation than pure W. It also exhibited low thermal conductivity at room temperature, but reached ≈120 W m −1 ·K −1 at ≈650 °C, surpassing pure W. This temperature‐dependent rise in κ, with a positive gradient of +0.075 W m −1 ·K − 2 , is attributed to decreasing diffuse mismatch at elevated temperatures. The large‐scale reaction and stable microstructure achieved through LMD process highlight its industrial potential. This work offers a strategy for developing high‐performance materials by combining RHEA's radiation resistance with Cu's thermal conductivity for extreme environments.

Chemistry↗
NASA NTRS2024

Expansion of Check-Cases for 6DOF Simulation

This is the Appendix containing a description of the solution for Case 1 in the assessment, “Expansion of Check-Cases for 6DOF Simulation”. For cases of spherical gravity, it is possible to provide a two-body solution without recourse to numerical integration and thus it is accurate to machine precision. Python code for a Keplerian Propagator (propagate.py) which produced a reference trajectory for Case 1 is provided in this appendix. There is also code for generating test cases which was used as an independent verification of the propagator. This is a high-level description of the algorithm employed. The documentation of each function includes implementation details, including equations for each task.

Modeling↗